Numerische Mathematik

, Volume 27, Issue 3, pp 257–269

A family of root finding methods

  • Eldon Hansen
  • Merrell Patrick

DOI: 10.1007/BF01396176

Cite this article as:
Hansen, E. & Patrick, M. Numer. Math. (1976) 27: 257. doi:10.1007/BF01396176


A one parameter family of iteration functions for finding roots is derived. The family includes the Laguerre, Halley, Ostrowski and Euler methods and, as a limiting case, Newton's method. All the methods of the family are cubically convergent for a simple root (except Newton's which is quadratically convergent). The superior behavior of Laguerre's method, when starting from a pointz for which |z| is large, is explained. It is shown that other methods of the family are superior if |z| is not large. It is also shown that a continuum of methods for the family exhibit global and monotonic convergence to roots of polynomials (and certain other functions) if all the roots are real.

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • Eldon Hansen
    • 1
  • Merrell Patrick
    • 2
  1. 1.Lockheed Palo Alto Research LaboratoryPalo AltoUSA
  2. 2.Computer Science DepartmentDuke UniversityDurhamUSA